Question

If A is a symmetric matrix and ${\text{n}} \in {\text{N}}$, write whether ${{\text{A}}^{\text{n}}}$is symmetric or skew-symmetric matrix or neither of these two.

Hint: As we know that A is symmetric matrix, i.e. ${\text{A = }}{{\text{A}}^{\text{T}}}$. So, taking power of n of A and then satisfying it in the known condition we can lead to the solution of the given statement.

Given, A is a symmetric matrix and ${\text{n}} \in {\text{N}}$,
As, A is a symmetric matrix, i.e. ${\text{A = }}{{\text{A}}^{\text{T}}}$.
Now, take transpose of ${{\text{A}}^{\text{n}}}$
$\Rightarrow {{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}$
As we know that ${{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = (}}{{\text{A}}^{\text{T}}}{{\text{)}}^{\text{n}}}$
$\Rightarrow {{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}}$
As we know that ${\text{A = }}{{\text{A}}^{\text{T}}}$
So, ${{\text{(}}{{\text{A}}^{\text{T}}}{\text{)}}^{\text{n}}} = {{\text{A}}^{\text{n}}}$
Hence, ${{\text{(}}{{\text{A}}^{\text{n}}}{\text{)}}^{\text{T}}}{\text{ = }}{{\text{A}}^{\text{n}}}$.
Hence, ${{\text{A}}^{\text{n}}}$is also a symmetric matrix.

Note: In mathematics, a matrix is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns. In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. In mathematics, particularly in linear algebra, a skew-symmetric matrix is a square matrix whose transpose equals it’s negative.

Additional information: Matrices can be used to compactly write and work with multiple linear equations, referred to as a system of linear equations, simultaneously. Matrices and matrix multiplication reveal their essential features when related to linear transformations, also known as linear maps.